Displacement gradient

Linear mapping of a vector by a tensor . ${\displaystyle {\vec {v}}}$ ${\displaystyle \mathbf {T} }$

In continuum mechanics, the displacement gradient (symbol:) is a unit-free tensor of the second level, which describes the local deformation in a material point of a body. Second level tensors are used here as linear mapping of geometric vectors onto geometric vectors, which are generally rotated and stretched in the process, see figure on the right. ${\displaystyle \mathbf {H} }$

The displacement of the particle of a body is the distance between its current position and its position in the (undeformed) starting position. The displacement gradient describes how the displacement changes if the position varies in the starting position. Mathematically, it is the gradient of the displacements associated vectors , hence the name. In the general case, the displacement gradient is dependent on both location and time. The components of the displacement gradient are calculated like a Jacobian matrix and can also be noted in a matrix .

The displacement gradient differs from the deformation gradient only in the constant unit tensor , but is mainly used in the case of small displacements. Small displacements exist when the largest displacements occurring in the body are still much smaller than a characteristic dimension of the body. In the case of small displacements, the displacement gradient is a fundamental variable with which local rotations, elongations and elongations are quantified. Its symmetrical component corresponds, for example, to the engineering expansion .

definition

The material body is mapped with configurations in a Euclidean vector space . In it the movement of a material point becomes with the movement function

${\displaystyle {\vec {x}}=x_{1}{\vec {e}}_{1}+x_{2}{\vec {e}}_{2}+x_{3}{\vec {e}}_{3}={\vec {\chi }}({\vec {X}},t)}$

described. The vector is the current position of the material point at the moment in the current configuration. The components are the spatial coordinates of the point with respect to the standard base . The vector ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {X}}}$ ${\displaystyle t}$ ${\displaystyle x_{1,2,3}}$ ${\displaystyle {\vec {e}}_{1,2,3}}$

${\displaystyle {\vec {X}}=X{\vec {e}}_{1}+Y{\vec {e}}_{2}+Z{\vec {e}}_{3}=X_{1}{\vec {e}}_{1}+X_{2}{\vec {e}}_{2}+X_{3}{\vec {e}}_{3}}$

is more precisely the position of the material point in question in the undeformed body in the initial or reference configuration. The components are the material coordinates of the point under consideration. ${\displaystyle X_{1,2,3}}$

With a fixed material point , the movement function describes its path through space. The displacement is now the difference vector between the current position of the point in the deformed body and its original position in the undeformed body: ${\displaystyle {\vec {X}}}$

${\displaystyle {\vec {u}}({\vec {X}},t)=u{\vec {e}}_{1}+v{\vec {e}}_{2}+w{\vec {e}}_{3}=u_{1}{\vec {e}}_{1}+u_{2}{\vec {e}}_{2}+u_{3}{\vec {e}}_{3}={\vec {\chi }}({\vec {X}},t)-{\vec {X}}={\vec {x}}-{\vec {X}}}$.

In order to investigate how the displacement changes when the position is varied in the undeformed starting position, the derivative is formed:

${\displaystyle H_{kl}={\frac {\mathrm {d} u_{k}({\vec {X}},t)}{\mathrm {d} X_{l}}}\quad {\mathsf {mit}}\quad k,l=1,2,3}$.

This contains the components of the displacement gradient with respect to the base system . ${\displaystyle H_{kl}}$ ${\displaystyle {\vec {e}}_{1,2,3}}$

To get a coordinate-free representation, the dyadic product is used: ${\displaystyle \otimes }$

${\displaystyle \mathbf {H} =\sum _{k,l=1}^{3}H_{kl}{\vec {e}}_{k}\otimes {\vec {e}}_{l}:=\operatorname {GRAD} \,{\vec {u}}({\vec {X}},t)={\begin{pmatrix}{\frac {\mathrm {d} u}{\mathrm {d} X}}&{\frac {\mathrm {d} u}{\mathrm {d} Y}}&{\frac {\mathrm {d} u}{\mathrm {d} Z}}\\{\frac {\mathrm {d} v}{\mathrm {d} X}}&{\frac {\mathrm {d} v}{\mathrm {d} Y}}&{\frac {\mathrm {d} v}{\mathrm {d} Z}}\\{\frac {\mathrm {d} w}{\mathrm {d} X}}&{\frac {\mathrm {d} w}{\mathrm {d} Y}}&{\frac {\mathrm {d} w}{\mathrm {d} Z}}\end{pmatrix}}_{{\vec {e}}_{k}\otimes {\vec {e}}_{l}}}$.

The tensor is the displacement gradient and is the symbol for the material gradient , because it is derived according to the material coordinates . ${\displaystyle \mathbf {H} }$ ${\displaystyle \operatorname {GRAD} }$ ${\displaystyle X_{1,2,3}}$

Geometric linearization

In solid mechanics , there are only small shifts in many areas, especially in technical areas. In this case, the equations of continuum mechanics experience a considerable simplification through geometric linearization . If is a characteristic dimension of the body, then both and are required for small displacements , so that all terms that contain higher powers of or can be neglected. The names for the deformation gradient ${\displaystyle L}$ ${\displaystyle |{\vec {u}}|\ll L}$ ${\displaystyle \parallel \mathbf {H} \parallel \ll 1}$ ${\displaystyle {\vec {u}}}$ ${\displaystyle \mathbf {H} }$

${\displaystyle \mathbf {F} =\operatorname {GRAD} {\vec {\chi }}=\operatorname {GRAD} {\vec {X}}+\operatorname {GRAD} {\vec {u}}=\mathbf {I} +\mathbf {H} }$,

the symmetrical

${\displaystyle \mathbf {H} ^{\mathrm {S} }={\frac {1}{2}}(\mathbf {H} +\mathbf {H} ^{\mathrm {T} })}$

and skew-symmetric part

${\displaystyle \mathbf {H} ^{\mathrm {A} }={\frac {1}{2}}(\mathbf {H} -\mathbf {H} ^{\mathrm {T} })}$

of the displacement gradient are used in the following. The linearized strain tensor

${\displaystyle {\boldsymbol {\varepsilon }}=\mathbf {H} ^{\mathrm {S} }={\frac {1}{2}}\left(\mathbf {H} +\mathbf {H} ^{\mathrm {T} }\right)={\begin{pmatrix}{\frac {\mathrm {d} u}{\mathrm {d} x}}&{\frac {1}{2}}\left({\frac {\mathrm {d} u}{\mathrm {d} y}}+{\frac {\mathrm {d} v}{\mathrm {d} x}}\right)&{\frac {1}{2}}\left({\frac {\mathrm {d} u}{\mathrm {d} z}}+{\frac {\mathrm {d} w}{\mathrm {d} x}}\right)\\{\frac {1}{2}}\left({\frac {\mathrm {d} v}{\mathrm {d} x}}+{\frac {\mathrm {d} u}{\mathrm {d} y}}\right)&{\frac {\mathrm {d} v}{\mathrm {d} y}}&{\frac {1}{2}}\left({\frac {\mathrm {d} v}{\mathrm {d} z}}+{\frac {\mathrm {d} w}{\mathrm {d} y}}\right)\\{\frac {1}{2}}\left({\frac {\mathrm {d} w}{\mathrm {d} x}}+{\frac {\mathrm {d} u}{\mathrm {d} z}}\right)&{\frac {1}{2}}\left({\frac {\mathrm {d} w}{\mathrm {d} y}}+{\frac {\mathrm {d} v}{\mathrm {d} z}}\right)&{\frac {\mathrm {d} w}{\mathrm {d} z}}\end{pmatrix}}}$

is well known in technical mechanics and is also called engineering stretching.

Deformation gradient and its polar decomposition

→ Main article : Deformation gradient

In the case of small displacements, the invariants of the deformation gradient are functions of the trace of the displacement gradient :

operator	general definition	Shape with small shifts
track	${\displaystyle \operatorname {Sp} (\mathbf {F} )=3+\operatorname {Sp} (\mathbf {H} )}$	${\displaystyle \operatorname {Sp} (\mathbf {F} )=3+\operatorname {Sp} (\mathbf {H} )}$
Second major invariant	${\displaystyle \operatorname {I} _{2}(\mathbf {F} )={\frac {1}{2}}(\operatorname {Sp} (\mathbf {F} )^{2}-\operatorname {Sp} (\mathbf {F\cdot F} ))}$	${\displaystyle \operatorname {I} _{2}(\mathbf {F} )\approx 3+2\operatorname {Sp} (\mathbf {H} )}$
Determinant	${\displaystyle \operatorname {det} (\mathbf {F} )=\operatorname {det} (\mathbf {I} +\mathbf {H} )}$	${\displaystyle \operatorname {det} (\mathbf {F} )\approx 1+\operatorname {Sp} (\mathbf {H} )}$
Frobenius norm	${\displaystyle \parallel \mathbf {F} \parallel ={\sqrt {\operatorname {Sp} (\mathbf {F^{\mathrm {T} }\cdot F} )}}}$	${\displaystyle \parallel \mathbf {F} \parallel \approx {\sqrt {3}}+\operatorname {Sp} (\mathbf {H} )}$

The deformation gradient can be clearly divided “polar” into a rotation and a pure stretching. The representation results from the application of the polar decomposition ${\displaystyle \mathbf {F} }$

${\displaystyle \mathbf {F} =\mathbf {R\cdot U} =\mathbf {v\cdot R} }$.

The rotation tensor is an “actually orthogonal tensor” . The material right stretch tensor and the spatial left stretch tensor are symmetrical and positive definite . In the case of small displacements, they are identical and linear in the linearized strains, as the following table shows: ${\displaystyle \mathbf {R} }$ ${\displaystyle \mathbf {U} }$ ${\displaystyle \mathbf {v} }$

Surname	general definition	Shape with small shifts
Right stretch tensor	${\displaystyle \mathbf {U} ={\sqrt {\mathbf {F^{\mathrm {T} }\cdot F} }}}$	${\displaystyle \mathbf {U} \approx \mathbf {I} +{\boldsymbol {\varepsilon }}}$
Left stretch tensor	${\displaystyle \mathbf {v} ={\sqrt {\mathbf {F\cdot F} ^{\mathrm {T} }}}}$	${\displaystyle \mathbf {v} \approx \mathbf {I} +{\boldsymbol {\varepsilon }}}$
Rotation tensor	${\displaystyle \mathbf {R} =\mathbf {F\cdot U} ^{-1}=\mathbf {v} ^{-1}\cdot \mathbf {F} }$	${\displaystyle \mathbf {R} \approx \mathbf {I} +\mathbf {H} ^{\mathrm {A} }}$

The identities

${\displaystyle {\begin{array}{lclclll}\mathbf {F} &=&\mathbf {R} &\cdot &\mathbf {U} \\\mathbf {H} &=&\mathbf {R} _{L}&+&{\boldsymbol {\varepsilon }}&=&\mathbf {H} ^{\mathrm {A} }+\mathbf {H} ^{\mathrm {S} }\end{array}}}$

show that with small distortions the polar decomposition of the deformation gradient changes into the additive decomposition of the displacement gradient in its skew-symmetrical and symmetrical part. The amount

${\displaystyle \mathbf {R} _{L}:=\mathbf {H} ^{\mathrm {A} }}$

becomes a linearized rotation tensor and the symmetric part

${\displaystyle {\boldsymbol {\varepsilon }}:=\mathbf {H} ^{\mathrm {S} }}$,

is called, as mentioned above, linearized strain tensor or engineering strain.

With the inverse of the tensors in the table, the sign of the portion of the displacement gradient is reversed in the case of geometric linearization:

${\displaystyle {\begin{array}{rcl}\mathbf {U} ^{-1}&=&\mathbf {v} ^{-1}\approx \mathbf {I} -{\boldsymbol {\varepsilon }}\\\mathbf {R} ^{-1}&=&\mathbf {R} ^{\mathrm {T} }\approx \mathbf {I} -\mathbf {H} ^{A}\end{array}}}$

Distance sensors

→ Main article : Stretch tensor

The right and left Cauchy-Green tensor are identical for small displacements and linear in the linearized strains:

Surname	general definition	Shape with small shifts
Right Cauchy-Green tensor	${\displaystyle \mathbf {C} =\mathbf {F^{\mathrm {T} }\cdot F} }$	${\displaystyle \mathbf {C} \approx \mathbf {I} +\mathbf {H} +\mathbf {H} ^{\mathrm {T} }=\mathbf {I} +2{\boldsymbol {\varepsilon }}}$
Left Cauchy-Green tensor	${\displaystyle \mathbf {b} =\mathbf {F\cdot F} ^{\mathrm {T} }}$	${\displaystyle \mathbf {b} \approx \mathbf {I} +\mathbf {H} +\mathbf {H} ^{\mathrm {T} }=\mathbf {I} +2{\boldsymbol {\varepsilon }}}$

Here too, when inverting in the geometrically linear case, the sign of is reversed: ${\displaystyle {\boldsymbol {\varepsilon }}}$

${\displaystyle \mathbf {C} ^{-1}\approx \mathbf {b} ^{-1}\approx \mathbf {I} -2{\boldsymbol {\varepsilon }}}$.

Strain tensors

→ Main article : strain tensor

With the above results for the distance tensor it can be confirmed immediately that the strain tensor changes into the linearized strain tensor or its negative one with small shifts : ${\displaystyle {\boldsymbol {\varepsilon }}}$

Surname	general definition	Shape with small shifts
Green-Lagrange strain tensor	${\displaystyle \mathbf {E} ={\frac {1}{2}}(\mathbf {C} -\mathbf {I} )}$	${\displaystyle \mathbf {E} \approx {\boldsymbol {\varepsilon }}}$
Biot strain tensor	${\displaystyle \mathbf {E} _{N}=\mathbf {U} -\mathbf {I} }$	${\displaystyle \mathbf {E} _{N}\approx {\boldsymbol {\varepsilon }}}$
Hencky stretches	${\displaystyle \mathbf {E} _{H}=\ln(\mathbf {U} )}$	${\displaystyle \mathbf {E} _{H}\approx \ln(\mathbf {I} +{\boldsymbol {\varepsilon }})\approx {\boldsymbol {\varepsilon }}}$
Piola strain tensor	${\displaystyle \mathbf {E} _{P}={\frac {1}{2}}(\mathbf {C} ^{-1}-\mathbf {I} )}$	${\displaystyle \mathbf {E} _{P}\approx -{\boldsymbol {\varepsilon }}}$
Euler-Almansi strain tensor	${\displaystyle \mathbf {e} ={\frac {1}{2}}(\mathbf {I} -\mathbf {b} ^{-1})}$	${\displaystyle \mathbf {e} \approx {\boldsymbol {\varepsilon }}}$
Finger tensor	${\displaystyle \mathbf {e} _{F}={\frac {1}{2}}(\mathbf {I} -\mathbf {b} )}$	${\displaystyle \mathbf {e} _{F}\approx -{\boldsymbol {\varepsilon }}}$
Swainger strain tensor	${\displaystyle \mathbf {e} _{S}=\mathbf {I} -\mathbf {v} ^{-1}}$	${\displaystyle \mathbf {e} _{S}\approx {\boldsymbol {\varepsilon }}}$.

See also

• Linearization
• Formula collection tensor algebra
• Tensoranalysis formula collection

Footnotes

1. ↑ ^{a b c d} The function value of a symmetrical, positively definite second-order tensor is calculated by means of its principal axis transformation , formation of the function value of the diagonal elements and inverse transformation.

literature

• H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 .
• P. Haupt: Continuum Mechanics and Theory of Materials . Springer, 2000, ISBN 3-540-66114-X .